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% AUC Accuracy Table
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\begin{figure}[!t]
\begin{center}
\begin{tabular}{|p{2cm}|p{1.5cm}|p{1cm}|} \hline
 Experiment & Accuracy & AUC \\ \hline
 Tcon & 0.972 & 0.699 \\ \hline
 Tpt & 0.976 & 0.703   \\    \hline 
 Svdpt & 0.9717 & 0.941  \\ \hline  
 Tcon-Svdpt & 0.9721 & 0.737\\  \hline
 Tpt-Svdpt & 0.976 & 0.825 \\ \hline 
\end{tabular}
\end{center}
\caption{\figtitle{Task1 AUC and Accuracy for various experiments.}
The table shows the accuracies resulting from training an SVM in 10-fold cross validation
for the various experiments with ``Tpt", ``Tcon" and ``Svdpt" in Task1.
}
\label{tab:task1_auc_tab}
\end{figure}

% Example of a figure

\begin{figure}[!t]
\begin{center}
\begin{tabular}{|p{2cm}|p{1.5cm}|p{1cm}|} \hline
 Experiment & Accuracy & AUC  \\\hline
 Tcon & 0.8707 & 0.935 \\\hline
 Tpt & 0.6522 & 0.743 \\\hline
 Svdpt & 0.6016 & 0.909 \\ \hline
 Tcon-Svdpt & 0.8730 & 0.936 \\\hline
 Tpt-Svdpt & 0.7003 & 0.863\\ \hline
\end{tabular}
\end{center}
\caption{\figtitle{Task2 AUC and Accuracy for various experiments.}
The table shows the accuracies resulting from training an SVM in 10-fold cross validation
for the various experiments with ``Tpt", ``Tcon" and ``Svdpt" in Task2.
}
\label{tab:task2_auc_tab}
\end{figure}

\section{Results of Experiments}
\label{sec:results}
We will use the terms similar to that described by E. Doi in \cite{doipaper}
to evaluate different model features. We describe the results of concatenating
features of Terrorist 
as ``Tcon", pointwise multiplication of features as
``Tpt". The pointwise multiplication of features from singular value decomposition
 is referred to as ``Svdpt". In the next few subsections, we describe the 
results of experiments where we use different combinations of these sets of 
features for both task 1 and 2. We use AUC as the primary
measure of performance, but also report $0/1$ accuracy.

\subsection{Task 1}
As explained in Section \ref{sec:experiments}, we perform experiments with different features and 
rank of matrix factorization. We evaluate our model using a 10-fold stratified
cross validation. 

\begin{figure}[!h]
\center
\includegraphics[width=0.5\textwidth]{img/Task1_rank_vs_AUC.eps}
\caption{\figtitle{AUC obtained for different ranks of matrix factorization
in Task1}
}
\label{fig:svdpt_k_task1}
\end{figure}

\paragraph{Variation of rank}
We evaluate our model to find the rank of matrix factorization that maximizes
AUC. We begin with the model ``Svdpt" and vary the rank to get the best
AUC. As shown in Figure \ref{fig:svdpt_k_task1}, we find that we obtain the highest AUC for rank $15$.
We use this rank for all the later experiments since the rank represents the top $k$
features from the Matrix factorization which are the most useful and remain 
the same for the rest of the experiments.  

\begin{figure}[!h]
\center
\includegraphics[width=0.5\textwidth]{img/Task1_expts.eps}
\caption{\figtitle{Different experiments and AUC, $0/1$ Accuracy obtained in Task1.}
Tpt-Svdpt performs the best. We do not choose Svdpt as our best model since the results
from that model are trivial with all negative predictions and no positive class predictions.
}
\label{fig:task1_auc}
\end{figure}

\begin{figure}[!t]
\begin{center}
\begin{tabular}{|p{2cm}|p{2cm}|p{2cm}|} \hline
  & True Negatives & True Positives\\ \hline
 Pred. Negatives & 28776 & 681\\ \hline
 Pred. Positives & 30 & 159 \\\hline
\end{tabular}
\end{center}
\caption{\figtitle{Confusion matrix for Tpt-Svdpt in Task1.}
Accuracy obtained is $0.976$.
}
\label{tab:task1_conf_mat}
\end{figure}

\paragraph{Combination of Features}
Figure \ref{fig:task1_auc} shows the AUC and $0/1$ accuracy for the various
feature combinations. We find that that ``Tpt-Svdpt" gives the best non-trivial 
AUC of \opttaskoneauc{} among all experiments. The corresponding accuracy obtained is
\opttaskoneacc{}. The confusion matrix for this result is shown in 
Figure \ref{tab:task1_conf_mat}. The complete results obtained for each
different experiment is shown in Figure \ref{tab:task1_auc_tab}.

\subsection{Task 2}
In task 2, we perform a similar set of experiments with different features as in task 1,
and evaluate our model through 10-fold stratified cross validation.

\begin{figure}[!h]
\center
\includegraphics[width=0.5\textwidth]{img/Task2_k_AUC.eps}
\caption{\figtitle{AUC obtained for different ranks of matrix factorization
in Task2}
}
\label{fig:svdpt_k_task2}
\end{figure}

\paragraph{Variation of rank}
Again, we start with the model ``Svdpt" and vary the rank of matrix factoriztion to get the highest
AUC. As shown in Figure \ref{fig:svdpt_k_task2}, we find that the rank \opttasktworank{} 
gave the best AUC. This result is similar to E. Doi's paper \cite{doipaper}.
We use this rank for all our subsequent experiments.

\begin{figure}[!h]
\center
\includegraphics[width=0.5\textwidth]{img/Task2_expts.eps}
\caption{\figtitle{Different experiments and AUC, $0/1$ Accuracy obtained in Task2.}
Tcon-Svdpt performs the best amongst all the models.
}
\label{fig:task2_auc}
\end{figure}

\begin{figure}[!t]
\begin{center}
\begin{tabular}{|p{2cm}|p{2cm}|p{2cm}|} \hline
  & Actual Negatives & Actual Positives \\\hline
 Pred. Negatives & 318 & 36 \\\hline
 Pred. Positives & 72 & 425 \\\hline
\end{tabular}
\end{center}
\caption{\figtitle{Confusion matrix for Tcon-Svdpt in Task2.}
Accuracy obtained is $0.873$.
}
\label{tab:task2_conf_mat}
\end{figure}

\paragraph{Combination of Features}
Figure \ref{fig:task2_auc} shows the AUC and $0/1$ accuracy for the various
feature combinations. We find that ``Tcon-Svdpt" gives the highest 
AUC of \opttasktwoauc{} among all experiments. The corresponding accuracy obtained is
\opttasktwoacc{}. The confusion matrix for this result is shown in 
Figure \ref{tab:task2_conf_mat}. The complete results obtained for each
different experiment is shown in Figure \ref{tab:task2_auc_tab}.

\subsection{Observations}
We observe that features from ``Svdpt" improve the performance of our models.
We attribute this to singular value decomposition capturing properties of
the network structure between the nodes, and thereby, the interaction between 
nodes. This aids in learning the model for both tasks.

The model using ``Tpt-Svdpt" performs best for task 1. We believe this is because
``Tpt" captures interaction between two nodes, which is important for
predicting the existence of a link. However, we also observe that ``Tcon-Svdpt" performs
the best for task 2, where we have to predict the type of the link. We
suspect that this is due to ``Tcon" capturing the individual node properties
which affect the type of a link more than ``Tpt" which captures interaction
between nodes. We think that propensity of two nodes contributes
more towards the nodes sharing the ``organizational" type of link, and this
when combined with ``Svdpt" yields the best model for task 2.
This result is similar to what is observed by E. Doi in his paper.

% Example of a table
\ignore {
\begin{figure*}[!t]
\begin{center}
\begin{tabular}{l l}
$- 0.099 *$ RFA\_2\_1 = 4 \\
$+ 0.110 *$ RFA\_2\_1 = 1 \\
\end{tabular}
\end{center}
\caption{\figtitle{Fast Large Margin Logistic Regression Model.}
The model produced by the linear SVM using the optimal cost value C found.}
\label{fig:logistic_model}
\end{figure*}
}
